LMFDB - Modular curve 120.360.13-30.e.1.2

Invariants

Level:	$120$	$\SL_2$-level:	$60$	Newform level:	$900$
Index:	$360$	$\PSL_2$-index:	$180$
Genus:	$13 = 1 + \frac{ 180 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps:	$6$ (none of which are rational)	Cusp widths	$30^{6}$	Cusp orbits	$2^{3}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$5 \le \gamma \le 24$
$\overline{\Q}$-gonality:	$5 \le \gamma \le 13$
Rational cusps:	$0$
Rational CM points:	none

Level structure

$\GL_2(\Z/120\Z)$-generators:	$\begin{bmatrix}29&52\\116&71\end{bmatrix}$, $\begin{bmatrix}39&10\\118&81\end{bmatrix}$, $\begin{bmatrix}61&110\\20&101\end{bmatrix}$, $\begin{bmatrix}63&22\\14&117\end{bmatrix}$, $\begin{bmatrix}67&80\\80&17\end{bmatrix}$, $\begin{bmatrix}85&102\\102&79\end{bmatrix}$, $\begin{bmatrix}109&2\\82&13\end{bmatrix}$, $\begin{bmatrix}117&40\\40&27\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 30.180.13.e.1 for the level structure with $-I$)
Cyclic 120-isogeny field degree:	$96$
Cyclic 120-torsion field degree:	$3072$
Full 120-torsion field degree:	$98304$

Models

Canonical model in $\mathbb{P}^{ 12 }$ defined by 55 equations

$ 0 $	$=$	$ v s + v d + s b - b d $
	$=$	$y c - y d - v a + a b$
	$=$	$y s - y c - t v - a b$
	$=$	$z s + z c - u v + u b$
	$=$	$\cdots$

[v*s + v*d + s*b - b*d, y*c - y*d - v*a + a*b, y*s - y*c - t*v - a*b, z*s + z*c - u*v + u*b, y*s + y*d + t*b + a*b, y*s + y*c + t*u + u*v + u*a - u*b, w*s + w*c + u*v + u*a, x*v + x*a + y*w, t*u - t*b - u*b + v*r + v*b - r*b, t*r - t*a + u*v + u*a + v*a + r*a, y*d + z*c - w*c - t*u - u*v + v^2 + v*r + v*b, y*d + z*c - w*c - t^2 - t*r + t*a - u*a + u*b, t*c - t*d - 2*s*a + a*c - a*d, x*u + 2*x*r + x*a - y*w - t*c - t*d, 2*x*t - 2*x*u + x*r + x*a + x*b - y*w - t*c - t*d + v*s - v*c + b*d, x*t - x*a - 2*x*b + y*w + t*c + t*d + 2*v*c + s*b - b*d, x*t - x*a - 2*x*b + y*w - t*c - v*c + s*a + a*d - b*c + b*d, x*u - 3*x*r + x*a - y*w + t*c - s*a - a*d, 2*x*u - 2*x*v - x*r + y*w + t*s + t*c - s*a, t*s + t*c + 2*u*c + v*s + v*c + r*s + r*c - s*a + s*b - a*c + b*c, t*c - t*d + u*s - u*c - 2*r*s - s*b + b*d, z*s - z*c + t*u - t*v - u*v - v^2 + v*a + v*b - a*b + b^2, y*d - z*s + w*c - t*u - t*v - t*a - u*b - r*a + a^2 - a*b, y*c + z*c + w*d - t*u - t*v - u*v + u*a + v*a + a*b, y*r - y*a + w*a + w*b, y*s + y*d + z*s - z*c + w*s - w*c + t*u - u*b, z*c - z*d + w*c - w*d + u*a - a*b, x*s - x*c + y*v - w*v + w*b, x*s - x*c - y*a + w*t + w*a, x*u + x*a - x*b - y*w + w^2 - t*d - u*d + v*c - r*s, x*r - x*a - z*w - w^2, x*u - x*r - x*a - 2*x*b + z^2 + t*c - u*d + v*c - r*s, x*v - x*b - y*z, x*t - x*b + y^2 - y*w, x*c - x*d - z*a, z*v + z*a + w*v - w*b, z*r - z*a + w*u - w*b, x*s + x*c - y*u, x*s - x*c + y*u - z*t - z*a, y*u - y*b - z*a - z*b, x*t + x*u + x*v - x*a + x*b - y*w + w^2, x*u + x*r - x*b - y*w - z*w - t*s + t*c - u*c + u*d - v*s + v*c - v*d - r*d - 2*b*d, x*y - x*z + 2*x*w - y*d - z*c + w*s + w*d - t*v + a*b, x^2 - y*t + y*v - z*u - w*u + s*c, x^2 - y*t + y*u - z*t + z*u - w*v - s*d, 2*x*y - 2*x*z - x*w + y*c - y*d - w*s + t*u, y*c + z*d - w*s + t^2 - t*u - t*v - t*r + u^2 - u*v - u*r + v^2 - v*r - r^2, z*u - 2*z*r + w*u - w*v + s*c + s*d - c^2 + d^2, y*t - 2*y*r - w*t + w*v - s*c + c^2 - 2*c*d, x^2 + y*t - y*v - y*r + z*v - z*r + w*u - w*v - w*b - s^2 - c*d + d^2, x^2 + y*t + z*u - z*v - w*t + s^2 - s*c + s*d, x^2 + x*s - x*c - x*d - y*u + y*r - z*u - z*r - w*r + w*a + s*c + s*d - c^2 + c*d, x^2 - x*c - y*v + z*v + z*b + w*r - w*b + s*d + d^2, 2*x^2 + x*d + z*t + z*r + 2*w*t + w*v - w*a - s*d + c*d - d^2, 3*x*y + 2*x*z + x*w - y*d - w*s - u*v]

Singular plane model Singular plane model

$ 0 $

$=$

$ - x^{8} y^{4} z^{2} + 8 x^{7} y^{6} z - 2 x^{7} y^{3} z^{4} + 4 x^{6} y^{8} + 19 x^{6} y^{5} z^{3} + \cdots + y^{2} z^{12} $

Rational points

This modular curve has no $\Q_p$ points for $p=13$, and therefore no rational points.

Maps to other modular curves

Map of degree 3 from the canonical model of this modular curve to the canonical model of the modular curve 10.60.3.a.1 :

$\displaystyle X$	$=$	$\displaystyle -y+z+3w$
$\displaystyle Y$	$=$	$\displaystyle 2y+3z-w$
$\displaystyle Z$	$=$	$\displaystyle 3y+2z+w$

Equation of the image curve:

$0$

$=$

$ 2X^{4}-3X^{3}Y-5X^{2}Y^{2}-4XY^{3}-2Y^{4}+3X^{3}Z-18X^{2}YZ-17XY^{2}Z+4Y^{3}Z-5X^{2}Z^{2}+17XYZ^{2}-6Y^{2}Z^{2}+4XZ^{3}+4YZ^{3}-2Z^{4} $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 30.180.13.e.1 :

$\displaystyle X$	$=$	$\displaystyle b$
$\displaystyle Y$	$=$	$\displaystyle x$
$\displaystyle Z$	$=$	$\displaystyle y$

Equation of the image curve:

$0$

$=$

$ -X^{8}Y^{4}Z^{2}+8X^{7}Y^{6}Z-2X^{7}Y^{3}Z^{4}+4X^{6}Y^{8}+19X^{6}Y^{5}Z^{3}+X^{6}Y^{2}Z^{6}+6X^{5}Y^{7}Z^{2}+50X^{5}Y^{4}Z^{5}+2X^{5}YZ^{8}-271X^{4}Y^{6}Z^{4}-35X^{4}Y^{3}Z^{7}-X^{4}Z^{10}-1439X^{3}Y^{8}Z^{3}-331X^{3}Y^{5}Z^{6}-18X^{3}Y^{2}Z^{9}-105X^{2}Y^{10}Z^{2}+2062X^{2}Y^{7}Z^{5}+261X^{2}Y^{4}Z^{8}+11X^{2}YZ^{11}+1300XY^{12}Z+3685XY^{9}Z^{4}+204XY^{6}Z^{7}-49XY^{3}Z^{10}+500Y^{14}+1700Y^{11}Z^{3}-354Y^{8}Z^{6}+128Y^{5}Z^{9}+Y^{2}Z^{12} $

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve	Level	Index	Degree	Genus	Rank
$X_{\mathrm{ns}}^+(3)$	$3$	$120$	$60$	$0$	$0$
40.120.3-10.a.1.2	$40$	$3$	$3$	$3$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
40.120.3-10.a.1.2	$40$	$3$	$3$	$3$	$0$
120.36.1-6.a.1.2	$120$	$10$	$10$	$1$	$?$

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